The only exception is it can not stream things you download from Amazon. Those can only be played on the main TiVo they're stored on.

Yea, I REALLY wish TiVo and/or Amazon would step this up a bit and make Amazon Prime streaming available on TiVos. The minis seem like the perfect opportunity for this too as they are pretty much strictly a streaming device. oh well

Probably not. Most premium channels are protected and as such can not be downloaded. Apparently they're working on a new feature to allow streaming outside the home, but there is some evidence it might also have restrictions on premium channels.